Our first observation about characters takes our work from last time and spins it in a new direction.
Let’s say and are conjugate elements of the group . That is, there is some so that . I say that for any -module with character , the character takes the same value on both and . Indeed, we find that
We see that is not so much a function on the group as it is a function on the set of conjugacy classes , since it takes the same value for any two elements in the same conjugacy class. We call such a complex-valued function on a group a “class function”. Clearly they form a vector space, and this vector space comes with a very nice basis: given a conjugacy class we define to be the function that takes the value for every element of and the value otherwise. Any class function is a linear combination of these , and so we conclude that the dimension of the space of class functions in is equal to the number of conjugacy classes in .
The basis isn’t orthonormal, but it is orthogonal. However, we can compute:
Incidentally, this is the reciprocal of the size of the centralizer of any . Thus if we pick a in each we can write down the orthonormal basis .